In this paper, we present a new computational framework using coupled and decoupled approximations for a Cahn-Hilliard-Navier-Stokes model with variable densities and degenerate mobility. In this sense, the coupled approximation is shown to conserve the mass of the fluid, preserve the point-wise bounds of the density and decrease an energy functional. In contrast, the decoupled scheme is presented as a more computationally efficient alternative but the discrete energy-decreasing property can not be assured. Both schemes are based on a finite element approximation for the Navier-Stokes fluid flow with discontinuous pressure and an upwind discontinuous Galerkin scheme for the Cahn-Hilliard part. Finally, several numerical experiments contrasting both approaches are conducted. In particular, results for a convergence test, a simple qualitative comparison and some well-known benchmark problems are shown.

翻译：本文针对具有变密度和退化迁移率的Cahn-Hilliard-Navier-Stokes模型，提出了一种使用耦合与解耦逼近的新计算框架。在此框架下，耦合逼近被证明能够保持流体的质量守恒、密度逐点有界性以及能量泛函的递减性。相比之下，解耦方法作为一种计算效率更高的替代方案被提出，但无法保证离散能量递减性质。两种方案均基于带间断压力的Navier-Stokes流体的有限元逼近，以及面向Cahn-Hilliard部分的上迎风间断Galerkin格式。最后，通过多次数值实验对比两种方法，展示了收敛性测试结果、简单定性比较以及若干经典基准问题的数值结果。